秩亏观测模型条件下的抗差Kalman滤波及其在组合导航中的应用

The robust Kalman filter, proposed in 1990s based on the generalized maximum likelihood estimator (M-estimator), has found widely applications in geodesy. However, how to evaluate quantitatively the robustness of the robust Kalman filter and its performance for rank deficient measurement models are still open problems. We are focused in this project in evaluating the robustness of the filtering method and robusify the Kalman filter for the rank deficient measurement models based on M-estimator. Then the corresponding theory is applied to the integrated navigation problem. The main contents are as follows. The non-Gaussian density function of the process and/or measurement noises is effectively represented by the Gauss mixture model. The measurement update of the Kalman filter is reformulated as a linear regression problem, and the rich theories in robust statistics can be easily implimented. The conceipt of the influence function of the Kalman filter is defined and derived, so the lack of robustness of the conventional Kalman filter is quantitatively revealed. The M-estimator, other than the least squares method is employed to solve the previously constructed linear regression problems to obtain the robust Kalman filter. In solving the M-estimator iteratively, how to choose iterating schemes and starting values is comprehensively studied in oder to account for the specific rank-dificiency problem. Finally, the analytical conclusions are applied in an important problem with rank-deficient measurement models, i.e., the integration of inertial navigation system and global navigation satellite system.

自20世纪90年代初提出以来，基于广义极大似然估计（M估计）的抗差Kalman滤波得到了广泛应用。然而，抗差Kalman滤波抗差性的定量评估及其在秩亏观测模型条件下的性能有待进一步的研究。本课题对滤波算法抗差性的定量评估以及秩亏观测模型条件下的基于M估计的Kalman滤波抗差化改进进行研究，并将研究成果用于组合导航系统，内容包括:用混合Gauss模型描述非Gauss分布；将Kalman滤波的观测更新过程表述为基于伪观测量的线性回归问题，以便采用成熟的抗差统计学理论对其进行研究；推导Kalman滤波的影响函数，定量揭示Kalman滤波的抗差性缺失问题；用M估计替代最小二乘估计，求解上述线性回归问题，以得到抗差Kalman滤波；对迭代求解M估中的迭代算法、初值选择等问题进行针对性研究；最后将研究结论用于INS/GNSS组合导航系统这一重要的、具有秩亏观测模型的滤波问题。

本课题对滤波算法抗差性的定量评估以及秩亏观测模型条件下的基于M估计的Kalman滤波抗差化改进进行研究，并将研究成果用于组合导航系统，内容包括:用混合Gauss模型描述非Gauss分布；将Kalman滤波的观测更新过程表述为基于伪观测量的线性回归问题，以便采用成熟的抗差统计学理论对其进行研究；推导Kalman滤波的影响函数，定量揭示Kalman滤波的抗差性缺失问题；用M估计替代最小二乘估计，求解上述线性回归问题，以得到抗差Kalman滤波；对迭代求解M估中的迭代算法、初值选择等问题进行针对性研究；最后将研究结论用于动态导航系统这一重要的、具有秩亏观测模型的滤波问题。